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Computable embeddings and strongly minimal theories

Published online by Cambridge University Press:  12 March 2014

J. Chisholm ,
J. F. Knight  and
S. Miller
J. Chisholm
Affiliation:
Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA, E-mail: JA-Chisholm@wiu.edu
J. F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556-5683. USA, E-mail: jlknight@nd.edu
S. Miller
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556-5683. USA, E-mail: smiller9@nd.edu
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Abstract

Here we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over ℚ, and ZS is the class of models of Th(ℤ, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 1031 - 1040
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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